Diffusion and consensus on weakly connected directed graphs

Let $G$ be a weakly connected directed graph with asymmetric graph Laplacian ${\cal L}$. Consensus and diffusion are dual dynamical processes defined on $G$ by $\dot x=-{\cal L}x$ for consensus and $\dot p=-p{\cal L}$ for diffusion. We consider both these processes as well their discrete time analogues. We define a basis of row vectors $\{\bar \gamma_i\}_{i=1}^k$ of the left null-space of ${\cal L}$ and a basis of column vectors $\{\gamma_i\}_{i=1}^k$ of the right null-space of ${\cal L}$ in terms of the partition of $G$ into strongly connected components. This allows for complete characterization of the asymptotic behavior of both diffusion and consensus --- discrete and continuous --- in terms of these eigenvectors. As an application of these ideas, we present a treatment of the pagerank algorithm that is dual to the usual one. We further show that the teleporting feature usually included in the algorithm is not strictly necessary. This is a complete and self-contained treatment of the asymptotics of consensus and diffusion on digraphs. Many of the ideas presented here can be found scattered in the literature, though mostly outside mainstream mathematics and not always with complete proofs. This paper seeks to remedy this by providing a compact and accessible survey.Comment: 19 pages, Survey Article, 1 figur

Social Network Analysis with sna

Modern social network analysis---the analysis of relational data arising from social systems---is a computationally intensive area of research. Here, we provide an overview of a software package which provides support for a range of network analytic functionality within the R statistical computing environment. General categories of currently supported functionality are described, and brief examples of package syntax and usage are shown.

Diàmetre unilateral en digrafs de doble pas

The main objective of this research work is to study the unilateral diameter in the (Delta,D) and (Delta,N) problems for double-step digraphs. A digraph is a network consisting of vertices and directed edges (called arcs). In the case of a graph, edges have no direction. A double-step digraph consists in a set of N vertices and arcs of the forms (i,i+a) and (i,i+b), with and a and b positive integers called "steps", that is, there exist arcs from the vertex i to the vertices i+a and i+b (all the operations are modulo N). This digraph is denoted by G(N;a,b). The diameter of a graph is the shortest path between two of the farthest vertices. In the diameter of a digraph, we must consider that the edges have directions. The unilateral diameter of a digraph is the minimum between the diameter (with directions) of the digraph and the diameter (with directions) of its converse digraph (obtained by changing the directions of all arcs). The (Delta,D) and (Delta,N) problems have been extensively studied for graphs and digraphs, but not in the case of double-step digraphs considering the unilateral diameter. In our context, the first problem is to find the maximum number of vertices N given an unilateral diameter D^* and a degree Delta=2, that is, to find out the two steps of a double-step digraph that maximize the number of vertices for these unilateral diameter and degree. The second problem is to find the minimum unilateral diameter D^* for a double-step digraph given a number of vertices N and a degree Delta=2, namely, to find out the two steps of a double- step digraph that minimize the unilateral diameter D^* for these number of vertices and degree.En termes generals, l'objectiu d'aquest treball és estudiar el diàmetre unilateral en els problemes i per al cas de digrafs de doble pas. Un digraf és una xarxa constituïda per vèrtexs i per arestes amb direcció (anomenades arcs). En el cas de grafs les arestes no tenen direcció. Un digraf de doble pas consta de vèrtexs i un conjunt d'arcs de la forma i , amb i enters positius anomenats "passos'', és a dir, que el vèrtex és adjacent cap als vèrtexs i (les operacions s'han d'entendre sempre mòdul ). Aquest digraf es denota . El diàmetre d'un graf és la mínima distància possible que hi ha entre dos dels vèrtexs més allunyats entre si. En el diàmetre d'un digraf hem de tenir en compte que els arcs tenen direcció. El diàmetre unilateral d'un digraf és el mínim entre el diàmetre (amb direccions) del digraf i el diàmetre (amb direccions) del seu digraf convers (obtingut canviant totes les direccions dels arcs). El problemes i han estat molt estudiats en grafs i en digrafs, però no en el cas dels digrafs de doble pas considerant el diàmetre unilateral. En el nostre context, el primer problema consisteix a trobar el màxim nombre de vèrtexs per a un diàmetre unilateral i el grau donats, és a dir, trobar quins són els dos passos d'un digraf de doble pas que fan que el nombre de vèrtexs sigui màxim per a aquests diàmetre unilateral i grau. El segon problema consisteix a trobar el mínim diàmetre unilateral en digrafs de doble pas per a un nombre de vèrtexs i el grau donats, és a dir, trobar quins són els dos passos d'un digraf de doble pas que fan que el diàmetre unilateral sigui mínim per a aquests nombre de vèrtexs i grau

Robust Observation and Control of Complex Networks

The problem of understanding when individual actions of interacting agents display to a coordinated collective behavior has receiving a considerable attention in many research fields. Especially in control engineering, distributed applications in cooperative environments are achieving resounding success, due to the large number of relevant applications, such as formation control, attitude synchronization tasks and cooperative applications in large-scale systems. Although those problems have been extensively studied in Literature, themost of classic approaches use to consider the unrealistic scenario in which networks always consist of identical, linear, time-invariant entities. It’s clear that this assumption strongly approximates the effective behavior of a network. In fact agents can be subjected to parameter uncertainties, unmodeled dynamics or simply characterized by proper nonlinear dynamics. Therefore, motivated by those practical problems, the present Thesis proposes various approaches for dealing with the problem of observation and control in both the framework of multi-agents and complex interconnected systems. The main contributions of this Thesis consist on the development of several algorithms based on concepts of discontinuous slidingmode control. This techniques can be employed for solving in finite-time problems of robust state estimation and consensus-based synchronization in network of heterogenous nonlinear systems subjected to unknown but bounded disturbances and sudden topological changes. Both directed and undirected topologies have been taken into account. It is worth to mention also the extension of the consensus problem to networks of agents governed by a class parabolic partial differential equation, for which, for the first time, a boundary-based robust local interaction protocol has been presented

Network analysis as applied to a group of AIDS patients linked by sexual contact

Thesis (B.S.) in Psychology -- University of Illinois at Urbana-Champaign, 1989.Includes bibliographical references (leaves 51-56).Microfiche of typescript. [Urbana, Ill.]: Photographic Services, University of Illinois, U of I Library, [1989]. 2 microfiches (83 frames): negative.s 1989 ilu n

Robust Observation and Control of Complex Networks

The problem of understanding when individual actions of interacting agents display to a coordinated collective behavior has receiving a considerable attention in many research fields. Especially in control engineering, distributed applications in cooperative environments are achieving resounding success, due to the large number of relevant applications, such as formation control, attitude synchronization tasks and cooperative applications in large-scale systems. Although those problems have been extensively studied in Literature, themost of classic approaches use to consider the unrealistic scenario in which networks always consist of identical, linear, time-invariant entities. It’s clear that this assumption strongly approximates the effective behavior of a network. In fact agents can be subjected to parameter uncertainties, unmodeled dynamics or simply characterized by proper nonlinear dynamics. Therefore, motivated by those practical problems, the present Thesis proposes various approaches for dealing with the problem of observation and control in both the framework of multi-agents and complex interconnected systems. The main contributions of this Thesis consist on the development of several algorithms based on concepts of discontinuous slidingmode control. This techniques can be employed for solving in finite-time problems of robust state estimation and consensus-based synchronization in network of heterogenous nonlinear systems subjected to unknown but bounded disturbances and sudden topological changes. Both directed and undirected topologies have been taken into account. It is worth to mention also the extension of the consensus problem to networks of agents governed by a class parabolic partial differential equation, for which, for the first time, a boundary-based robust local interaction protocol has been presented

Social Network Analysis with sna

Modern social network analysis---the analysis of relational data arising from social systems---is a computationally intensive area of research. Here, we provide an overview of a software package which provides support for a range of network analytic functionality within the R statistical computing environment. General categories of currently supported functionality are described, and brief examples of package syntax and usage are shown